MinT - Best Known (162−53, 162, s)-Nets in Base 3

Best Known (162−53, 162, s)-Nets in Base 3

(162−53, 162, 156)-Net over F₃ — Constructive and digital

Digital (109, 162, 156)-net over F₃, using

12 times m-reduction [i] based on digital (109, 174, 156)-net over F₃, using
- trace code for nets [i] based on digital (22, 87, 78)-net over F₉, using
  - net from sequence [i] based on digital (22, 77)-sequence over F₉, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₉ with g(F) = 22 and N(F) ≥ 78, using
      - F₃ from the tower of function fields by GarcÃa and Stichtenoth over F₉ [i]

(162−53, 162, 283)-Net over F₃ — Digital

Digital (109, 162, 283)-net over F₃, using

Gilbertâ€“VarÅ¡amov bound [i]

(162−53, 162, 4725)-Net in Base 3 — Upper bound on s

There is no (109, 162, 4726)-net in base 3, because

1 times m-reduction [i] would yield (109, 161, 4726)-net in base 3, but
- the generalized Rao bound for nets shows that 3^mÂ â‰¥ 65860 399848 053015 319356 251173 493464 196946 263271 187356 470031 338370 069509 786173 > 3¹⁶¹ [i]